Consider a $(D\times D)$ symmetric matrix $\sfA$ whose entries are linear forms in $\Q[X_1, \ldots, X_k]$ with coefficients of bit size $\leq \tau$. We provide an algorithm which decides the existence of rational solutions to the linear matrix inequality $\sfA\succeq 0$ and outputs such a rational solution if it exists. This problem is of first importance: it can be used to compute algebraic certificates of positivity for multivariate polynomials. Our algorithm runs within $(k\tau)^{O(1)}2^{O(\min(k, D)D^2)}D^{O(D^2)}$ bit operations; the bit size of the output solution is dominated by $\tau^{O(1)}2^{O(\min(k, D)D^2)}$. These results are obtained by designing algorithmic variants of constructions introduced by Klep and Schweighofer. This leads to the best complexity bounds for deciding the existence of sums of squares with rational coefficients of a given polynomial. We have implemented the algorithm; it has been able to tackle Scheiderer's example of a multivariate polynomial that is a sum of squares over the reals but not over the rationals; providing the first computer validation of this counter-example to Sturmfels' conjecture.